On the Center of the Brauer Algebra

Shalile, Armin

doi:10.1007/s10468-011-9294-2

Cited by 8 publications

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References 16 publications

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“…In particular, we have dim Z(B 2,2 (δ)) = 6, which is independent from δ (see also the conjecture on the dimension of the center of the Brauer algebra in [35,Introduction]).…”

Section: Blocks Of Walled Brauer Algebramentioning

confidence: 85%

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Supersymmetric polynomials and the center of the walled Brauer algebra

Jung,

Kim

2015

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We study a commuting family of elements of the walled Brauer algebra Br,s(δ), called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When Br,s(δ) is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption, we define a maximal commutative subalgebra of Br,s(δ), called the Gelfand-Zetlin subalgebra, and show that it is generated by the Jucys-Murphy elements. As an application, we construct a complete set of primitive orthogonal idempotents of Br,s(δ), when it is semisimple. We also give an alternative proof of a part of the classification theorem of blocks of Br,s(δ) in non-semisimple cases, which appeared in the work of Cox-De Visscher-Doty-Martin. Finally, we present an analogue of Jucys-Murpy elements for the quantized walled Brauer algebra Hr,s(q, ρ) over C(q, ρ) and by taking the classical limit we show that the supersymmetric polynomials in these elements generates the center. It follows that H. Morton conjecture, which appeared in the study of the relation between the framed HOM-FLY skein on the annulus and that on the rectangle with designated boundary points, holds if we extend the scalar from Z[q ±1 , ρ ±1 ] (q−q −1 ) to C(q, ρ).

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“…In particular, we have dim Z(B 2,2 (δ)) = 6, which is independent from δ (see also the conjecture on the dimension of the center of the Brauer algebra in [35,Introduction]).…”

Section: Blocks Of Walled Brauer Algebramentioning

confidence: 85%

“…Each element of the basis corresponds to a walled generalized cycle types. For the detail, see [35,Section 7]. We enumerate thus obtained elements as follows:…”

Section: Blocks Of Walled Brauer Algebramentioning

confidence: 99%

Supersymmetric polynomials and the center of the walled Brauer algebra

Jung,

Kim

2015

Preprint

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“…In particular, we are able to provide an explicit algorithm for the determination of a basis of the center for all but finitely many values of the parameter, see [22].…”

Section: Introductionmentioning

confidence: 99%

Conjugation in Brauer algebras and applications to character theory

Shalile¹

2011

Journal of Pure and Applied Algebra

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a b s t r a c tThe Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.In particular, there is an ordering of simple and cell modules such that D B r (δ) is lower triangular with diagonal entries equal to 1. Remark 1.10. When F is a field of characteristic 0, then the decomposition matrix of the Brauer algebra was determined by Martin, see [16]. If the characteristic of F is 0 or larger than p, then an algorithm to compute the decomposition matrix of the Brauer algebra can also be found in [23]

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“…We remark that, similar to the case of finite groups, the knowledge of conjugacy classes gives a lot of information about the center of the Brauer algebra. In particular, we are able to provide an explicit algorithm for the determination of a basis of the center for all but finitely many values of the parameter, see [Sh09].…”

Section: Introductionmentioning

confidence: 99%

Conjugation in Brauer algebras and applications to character theory

Shalile

2009

Preprint

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The Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define modular characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of ordinary characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.

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On the Center of the Brauer Algebra

Cited by 8 publications

References 16 publications

Supersymmetric polynomials and the center of the walled Brauer algebra

Supersymmetric polynomials and the center of the walled Brauer algebra

Conjugation in Brauer algebras and applications to character theory

Conjugation in Brauer algebras and applications to character theory

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